The answer to an email of Mr. Douglas C. Ravenel Zhou Xueguang Department of Mathematics, Nankai University Tianjin 300071, China October 27, 2003 Due to a mistake I made myself, I announce to withdraw the article Ä Reply"* * that appeared on the September 2003 submissions. The following is the answer to an email of Mr Douglas C. Ravenel on January * *12,2002. I am sorry for replying so late. (1) In your email, you said that the Serre spectral sequence of the followin* *g sequence ! (S2m+1 ) ! (B(mp)) ! (S2m+2p+1) ! collapse and so we have the isomorphism H*( (B(mp))) = H*( (S2m+1 ) H*( (S2m+2p* *+1)) as an additive group, but we do not know whether this is a ring isomorphism. (2) In the second part, you said that you do not believe that as maps from M* *(l+m+ n+2) to M(l)^M(m)^M(n), P ((ff fi) fl) = fl (ff fi) and id((ff fi) fl) = (ff fi* *) fl are homotopic. I agree that it needs a proof. Now, we will give a proof in the * *last part of this letter. The other part of the appendix on page 48 of my paper remain uncha* *nged. So, I still believe that the two proofs of Toda's result ff1fip1= 0 for p > 3 a* *re incorrect. (3) Besides above, there is still a gap in the proof of the statement that V* * (3) does not exist for n > 5. In page 290 of your book öC mplex Cobordism and Stable Homotopy Groups of Spheres", you said that x761=< ff1fi3, fi4, fl2 > is a permanent cycl* *e for p = 5. But we have |ff1| = 7, |fi3| = 134, |fi4| = 82, |fl2| = 437, and so the Massey * *product x761 exists modular a coset ufl2 + (ff1fi3)v with |u| = 324 and |v| = 620. From the * *table in page 291 we know that u and v may not be trivial. Then, how can you get the con* *clusion that x761is a permanent cycle? (4) In what follows, we will prove that (ff fi) fl ff (fi fl). the* * Moore space M(l), M(m) and M(n) are the same as defined in my paper. Since ff (fi fl) 1 (-1)(l+1)(m+n+2)(fi fl) ff, if m = n, then ff (fi fl) (fi fl) ff.* * So, we have P ((ff fi) fl) (fi fl) ff ff (fi fl) (ff fi) fl for m = n* * = l. Firstly, we introduce the notions that will be used. It is assumed that all* * maps between spaces with base point keep the base point. For two spaces X, Y , X = Y means that there is a given topological map f: X ! Y . Let n > 1. As usual, we use Dn to denote the unit disk in Rn and Sn to denote the unit sphere in Rn+1. * * So, @(Dn+1) = Sn. The base point ] of Dn and Sn-1 is taken to be (-1, 0, . .,.0). * * It is obvious that there exists a topological map H from Sn to ^ni=1S1i, where S1iis * *just a copy of S1. We say that H is a decomposition of Sn. In general, we omit the map* * H and use the notion Sn = ^ni=1S1ito denote it. It is easy to check that Dn ^ D1 * *= Dn+1. By induction, we have Dn ^ Dl = Dn+l for n, l > 1. It is also obvious that Dn+* *1 = @(Dn+1) ^ D1 = Sn ^ D1. By the decomposition @(Dn+1) = Sn = ^ni=1S1i, any point* * x of Dn+1 can be expressed as x1 ^ . .^.xn ^ t with xi2 S1i, i = 1, . .,.n and -1* * 6 t 6 1. We say that x1, . .,.xn, t is the polar coordinates of x with respect to the de* *composition Sn = ^ni=1S1iand Dn+1 = Sn ^ D1. We also have that Dn+1 = (^ni=1S1i) ^ D1 is a decomposition of Dn+1. In this appendix, p is always assumed to be a fixed odd * *prime. Let `: S1 ! S1 be defined by `(e2iix) = e2iipx. It is obvious that ` has degre* *e p and we call it the standard p-map of S1. Let `n: Sn ! Sn be the map defined by that `n(x1^ . .^.xn) = `(x1) ^ . .^.xn for all xi2 S1i, i = 1, . .,.n. We call `n th* *e standard p-map of Sn with respect to the decomposition Sn = ^ni=1S1i. The identification* * space M~(n) = Dn+1 [ Sn=x ^ 1 in Dn+1 are identified with `n(x) 2 Sn for all x 2 Sn a* *nd M~(n) = Dn+1 [ Sn x (-1, 1]=x ^ t in Dn+1 are identified with (`n(x), t) for al* *l x 2 Sn and -1 < t 6 1 are respectively call S Moore space and SS Moore space. It is ob* *vious that M~(n) M~(n).We call Dn+1 the standard (n + 1)-cell of M~(n) and M~(n) an* *d call Sn the unit sphere of them. We use ~n+1 to denote both inclusion maps from Dn+1* * to M~(n) and M~(n). Notice that in M~(n) and M~(n), @Dn+1 and the unit sphere Sn are not the same space. For x 2 @Dn+1, we have x 62 `n(x) 2 Sn. In M~(n), any point can be expre* *ssed as x ^ t with x 2 Sn and -1 6 t 6 1. For l > 0, m > 0, let Dl+1= Sl^D11= (^li=1)^D11and Dm+1 = Sm ^D12= (^mi=1)^D* *12 __ be two decompositions of Dl+1 and Dm+1 . Let D2 = D1 ^ D2 = (@D2) ^ D 1, then * * __ Dl+1^ Dm+1 = (Sl^ D11) ^ (Sm ^ D12) = Sl^ Sm ^ (D11^ D12) = (^l+m+1i=1) ^ (@D2)* * ^ D1. 2 Thus, we get a decomposition of Dl+1^ Dm+1 . We call this the natural decomposi* *tion of Dl+1^ Dm+1 with respect to the decomposition of the factors Dl+1 and Dm+1 . Let M~(l) and M~(m) be two SS Moore spaces with respectively standard cell D* *l+1 and Dm+1 and standard spheres Sl and Sm . Then, any point of M~(l) ^ M~(m) can * *be expressed as (x, t1) ^ (y, t2) with x 2 Sl, y 2 Sm and -1 6 t1, t2 6 1. We wr* *ite it as (x ^ y ^ (t1, t2). By the definition of M~(l) and M~(m), we have x ^ t = (`l(x* *), t) and y^t = (`m (y), t) for x 2 @Dl+1and y 2 @Dm+1 and -1 < t 6 1. Now, @(Dl+1^Dm+1 )* * = {x ^ t1 ^ y ^ t2 | x 2 Sl, y 2 Sm , t1 = 1, -1 6 t2 6 1 or t2 = 1, -1 6 t1 6 1}* *. So, we have the following Proposition 1. For x 2 @Dl+1, y 2 @Dm+1 , t1 = 1, -1 6 t2 6 1 or t2 = 1, -1 * *6 t1 6 1, ~l^ ~m (x ^ t1 ^ y ^ t2) = (`l(x), t1) ^ (`m (y), t2). Let M~(l + m + 1) and M~(l + m + 1) be the S and SS Moore spaces defined abo* *ve. We defined a map ~ 0(l, m): ~M(l+m+1) ! M~(l)^M~ (m) as follows. For x 2 Dl+1, y 2* * Dm+1 , ~ 0(l, m)(Dl+m+1) = ~l(x) ^ ~m (y). Since Sl+m+1 = {x ^ t1^ y ^ t2 | x 2 Sl, y * *2 Sm , t1 = 1, -1 6 t2 6 1 ort2 = 1, -1 6 t1 6 1}, we define a map ~ 00(l, m): Sl+m+1 ! M~(* *l)^M~ (m) as follows. For x 2 Sl and y 2 Sm , ~ 00(l, m)(x ^ t1 ^ y ^ t2) = x ^ `m (y) ^ * *(t1, t2) for t2 = 1, -1 6 t1 6 1 and t2 = 1, -1 6 t1 6 1. Let `l+m+1 be the p-map defined by the decomposition Dl+m+2 = Dl+1^ Dm+1 with respect to the factors Dl+1and Dm+1 . By Proposition 1, we have ~ 0(l, m)(x ^ y) = ~ 00(l, m)((`(x ^ y)) for x ^ y 2 * *@Dl+m+2. So, ~ 0(l, m) and ~ 00(l, m) define a map ~(l, m): ~M(l + m) ! M~(l) ^ M(m). We* * extend ~ (l, m) to a map ~ (l, m): ~M(l + m + 1) ! M~(l) ^ M(m) as follows. For -1 6 t* * 6 1, let jt: [-1, 1] ! [-1, t] be the linear map such that jt = -1 and jt(1) = t. Le* *t Dl+1t= Sl^ (-1, t], Dm+1t= Sm ^ (-1, t]. Define M~t(l) = Dl+1t^ Slx t= x ^ t are ident* *ified with `l(xxt for all x 2 @Dl+1. M~t(m) is similarly defined. We have M~(l) = [t2(-1,1* *]~Mt(l) and M~(m) = [t2(-1,1]~Mt(m). Let jlt: ~M(l) ! M~t(l) and jmt: ~M(m) ! M~t(m) be res* *pectively the maps defined by jlt(x ^ ø) = x ^ jt(ø) for x 2 @Dl+1and jlt(x0^ 1) = (x, t)* * for x02 Sl and jmt(y ^ ø) = y ^ jt(ø) for y 2 @Dm+1 and jmt(y0^ 1) = (y, t) for y02 Sm . W* *e extend ~ (l, m) to a map e~(l, m): ~M(l + m + 1) ! M~(l + m + 1) [ Sl+m+1 x (-1, 1] as* * follows. For x 2 Sl+m+1 and -1 6 t 6 1, e~(l, m)(x, t) = (jlt^ jmt) O ~(l, m)(x, 1) 2 M~* *lt^ M~mt M~(l) ^ M~(m). It can be easily seen that the map e~(l, m) so defined is contin* *uous and e~(l, m) l+m+1 l m+1 l+1 l+1 m ~ l+1 m+1 *(S ) = S ^D +(-1) D ^S and (l, m)*(Dl+m+2) = D ^D . So, both e~(l, m) and ~(l, m) are ff fi maps. 3 Now, we prove the main proposition of this appendix (ff fi) fl = ff (f* *i fl). Let ff = M~(l), fi = M~(m), fl = M~(n) be three SS Moore spaces and Dl+1, Dm+1 , Dn* *+1 be respectively the standard cells of ff, fi and fl. Dl+1= (^li=1S1i)^D11, Dm+1 = * *(^li=mS1i)^ D12, Dn+1 = (^li=nS1i) ^ D13. Now, there are two different decompositions of Dl* *+m+n+3. The first is Dl+m+n+3 = Dl+m+2 ^ Dn+1 and the second is Dl+m+n+3 = Dl+1^ Dm+n+2* * . Let `(1)l+m+n+2and `(2)l+m+n+2be respectively the standard p-maps with the firs* *t and second decomposition. Then, we have the following Proposition 2. For all x 2 Sl+m+n+2 , `(1)l+m+n+2(x) = `(2)l+m+n+2(x), and * *there exist two disks ~D1and ~D1such that D11^ D12= @(D11^ D12) ^ ~D1, D12^ D13= @(D12^ D13) ^ ~D2 Proof. We have Dl+m+2 = Dl+1^ Dm+1 = (^l+mi=1S1i) ^ (D11^ D12) = (^l+mi=1S* *1i) ^ @(D11^ D12) ^ ~D1and Dl+m+n+3 = Dl+m+2 ^ Dn+1 = (^l+mi=1S1i) ^ @(D11^ D12) ^ ~D* *1^ D13. It can be easily seen that @(D11^ D12) ^ @(D12^ D13) = @(D11^ D12^ D13). Since* * any point x in Sl+m+n+2 can be expressed by ^l+m+ni=1xi^ T1 ^ T2 with T1 2 @(D11^ D* *12) and T1 2 @(D~11^ D13), we have by definition `(1)l+m+n+2(x) = `(x1) ^ (^l+m+ni=2xi)* * ^ T1 ^ T2. Since any point T in @(D11^ D12^ D13) can be expressed by T1 ^ T2 and any point* * x can be expressed by (^l+m+ni=1S1i) ^ T , we have `(1)l+m+n+2(x) = `(x1) ^ (^l+m+ni=* *2xi) ^ T . By the same method as above, we also have `(2)l+m+n+2(x) = `(x1) ^ (^l+m+ni=2xi) ^* * T . So, we have `(1)l+m+n+2(x) = `(2)l+m+n+2(x). Proposition 3. (l, m) ^ id) O (l + m + 1, n) = id^ (m, n)) O (l, m + n +* * 1). Proof. Notice that in the definition of ~(l + m + 1, n) and ~(l, m + n + 1),* * the two standard p-maps are the same, so the two M~(l + m + n + 1 are the same. Now, we* * have the following commutative diagram @(Dl+m+n+3) -id! @(Dl+m+2^ Dn+1) -id! @(Dl+1^Dm+1 ^Dn+1) # ~`l+m+n+3 # ~l+m ^ ~n # ~l^ ~m ^ ~n Sl+m+n+2 - ! ~M(l+ m+ 1)^ ~M(n) -! M~(l)^ ~M(m)^ ~M(n) where the two maps on the bottom line are respectively ~(l+ m+ 1, n) and ~(l+ m* *+ 1, n)^ id. For x 2 Sl+m+n+2 , there exists an x0 2 @Dl+m+n such that x = `(1)l+m+n+2(* *x0) = `(2)l+m+n+2(x0). Since (~ (l, m)^ id)(~ (l + m + 1, n))(x) = (~ (l, m)^ id)(~ (l + m + 1, n))~`l+m+n+2(x0) 4 = (~l^~m ^~n)(id)(id)(x0) = (~l^~m ^~n)(x0) Similarly, we have (id^ ~(m, n))(~ (l, m + n + 1))(x) = (~l^~m ^~n)(x0). So, w* *e have (k = l + m + n + 2) ( (l, m)^ id)(~ (l + m + 1, n))|Sk = (id^ ~(m, n))(~ (l, m + n + 1))|Sk* * . Since both maps on Dl+m+n+3 are ~l^~m ^~n, we have ( (l, m)^ id)(~ (l + m + 1, n)) = id^(~ (m, n))(~ (l, m + n + 1)). 5